Question

care dintre numerele naturale x și y este mai mic x egal cu 4 la 60 minus 4 la 58 minus 4 la 57 și y egal cu 3 la 118 minus 3 la 117 plus 3 la 116 minus 3 la 115 minus 3 la 114​

Answer 1

Răspuns:

x

Explicație pas cu pas:

$x=4^{60}-4^{58}-4^{57};\\\\x=(2^2)^{60}-(2^2)^{58}-(2^2)^{57};\\\\x=2^{2*60}-2^{2*58}-2^{2*57}\\\\x=2^{120}-2^{116}-2^{114}\\\\x=2^{114}(2^{120-114}-2^{116-144}-2^{114-144})\\\\x=2^{114}(2^{6}-2^{2}-2^{0})\\\\x=2^{114}(64-4-1)=2^{114}*59;\\\\y=3^{118}-3^{117}+3^{116}-3^{115}-3^{114}\\\\y=3^{114}(3^{118-114}-3^{117-114}+3^{116-114}-3^{115-114}-3^{114-114})\\\\y=3^{114}(3^4-3^3+3^2-3^1-3^0)\\\\y=3^{114}(81-27+9-3-1)\\\\y=3^{114}*59$

$Acum\ le\ putem\ compara\\\\x=2^{114}*59\ \ \ si \ \ \ y=3^{114}*59\\\\59\ este\ comun\ la\ ambele\ deci\ comparam\ 2^{114}\ cu\ 3^{114}\\\\Ambii\ au\ acelasi\ exponent\ deci\ comparam\ bazele.\\\\2\ este\ mai\ mic\ decat\ 3,\ deci\ cel\ mai\ mic\ dintre\ x\ si\ y\ este\ x;$